Abstract
We study the harvesting of correlations by two Unruh-DeWitt static detectors from the vacuum state of a massless scalar field in a background Vaidya spacetime consisting of a collapsing null shell that forms a Schwarzschild black hole (hereafter Vaidya spacetime for brevity), and we compare the results with those associated with the three preferred vacua (Boulware, Unruh, Hartle-Hawking-Israel vacua) of the eternal Schwarzschild black hole spacetime. To do this we make use of the explicit Wightman functions for a massless scalar field available in (1+1)-dimensional models of the collapsing spacetime and Schwarzschild spacetimes, and the detectors couple to the proper time derivative of the field. First we find that, with respect to the harvesting protocol, the Unruh vacuum agrees very well with the Vaidya vacuum near the horizon even for finite-time interactions. Second, all four vacua have different capacities for creating correlations between the detectors, with the Vaidya vacuum interpolating between the Unruh vacuum near the horizon and the Boulware vacuum far from the horizon. Third, we show that the black hole horizon inhibits any correlations, not just entanglement. Finally, we show that the efficiency of the harvesting protocol depend strongly on the signalling ability of the detectors, which is highly non-trivial in presence of curvature. We provide an asymptotic analysis of the Vaidya vacuum to clarify the relationship between the Boulware/Unruh interpolation and the near/far from horizon and early/late-time limits. We demonstrate a straightforward implementation of numerical contour integration to perform all the calculations.
Highlights
The pioneering work of Valentini [10] and later by Reznik et al [11, 12] showed that one can extract entanglement from the QFT vacuum using a pair of initially uncorrelated quantum systems: in recent years this protocol became known as entanglement harvesting
We study the harvesting of correlations by two Unruh-DeWitt static detectors from the vacuum state of a massless scalar field in a background Vaidya spacetime consisting of a collapsing null shell that forms a Schwarzschild black hole, and we compare the results with those associated with the three preferred vacua (Boulware, Unruh, Hartle-Hawking-Israel vacua) of the eternal Schwarzschild black hole spacetime
It was found that the Unruh vacuum gives an upper bound for radiation flux at future null infinity, and a stationary observer carrying a detector at fixed orbit in the exterior black hole region measures a Planckian spectrum in the late time limit [39]
Summary
One can use ingoing Eddington-Finkelstein coordinates (U, v, θ, φ) defined by V = ev/(4M), with v ∈ R In this coordinate system, we can view regions I and II as an asymptotically flat and globally hyperbolic spacetime in itself, denoted (ME, gE), where ME is a submanifold of MK and gE is the induced metric obtained from inclusion map i : ME → MK by pullback, i.e. gE = i∗gK. We can view regions I and II as an asymptotically flat and globally hyperbolic spacetime in itself, denoted (ME, gE), where ME is a submanifold of MK and gE is the induced metric obtained from inclusion map i : ME → MK by pullback, i.e. gE = i∗gK In this coordinate system, the metric reads gE.
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